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Properties

Label 287.3.d.b
Level $287$
Weight $3$
Character orbit 287.d
Self dual yes
Analytic conductor $7.820$
Analytic rank $0$
Dimension $7$
CM discriminant -287
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [287,3,Mod(286,287)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(287, base_ring=CyclotomicField(2))
 
chi = DirichletCharacter(H, H._module([1, 1]))
 
N = Newforms(chi, 3, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("287.286");
 
S:= CuspForms(chi, 3);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 287 = 7 \cdot 41 \)
Weight: \( k \) \(=\) \( 3 \)
Character orbit: \([\chi]\) \(=\) 287.d (of order \(2\), degree \(1\), minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(7.82018358714\)
Analytic rank: \(0\)
Dimension: \(7\)
Coefficient field: 7.7.19468476636329.1
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{7} - 14x^{5} + 56x^{3} - 56x - 15 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2, a_3]\)
Coefficient ring index: \( 1 \)
Twist minimal: yes
Sato-Tate group: $\mathrm{U}(1)[D_{2}]$

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{6}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_{2} q^{2} - \beta_{4} q^{3} + (\beta_{4} + \beta_{3} + 4) q^{4} + ( - \beta_{6} + \beta_{5} - 2 \beta_{2}) q^{6} + 7 q^{7} + (\beta_{6} + 4 \beta_{2} + 4 \beta_1) q^{8} + ( - \beta_{6} - 5 \beta_1 + 9) q^{9}+O(q^{10}) \) Copy content Toggle raw display \( q + \beta_{2} q^{2} - \beta_{4} q^{3} + (\beta_{4} + \beta_{3} + 4) q^{4} + ( - \beta_{6} + \beta_{5} - 2 \beta_{2}) q^{6} + 7 q^{7} + (\beta_{6} + 4 \beta_{2} + 4 \beta_1) q^{8} + ( - \beta_{6} - 5 \beta_1 + 9) q^{9} + (2 \beta_{6} - 4 \beta_{4} + \beta_1 - 17) q^{12} + ( - 3 \beta_{5} + \beta_{2}) q^{13} + 7 \beta_{2} q^{14} + ( - 2 \beta_{6} + 4 \beta_{4} + \cdots + 16) q^{16}+ \cdots + 49 \beta_{2} q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 7 q + 28 q^{4} + 49 q^{7} + 63 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 7 q + 28 q^{4} + 49 q^{7} + 63 q^{9} - 119 q^{12} + 112 q^{16} + 175 q^{25} + 77 q^{26} + 196 q^{28} + 252 q^{36} - 287 q^{41} - 476 q^{48} + 343 q^{49} + 469 q^{54} + 441 q^{63} + 448 q^{64} - 182 q^{69} - 1001 q^{72} - 973 q^{74} - 917 q^{78} + 567 q^{81} - 833 q^{84} - 721 q^{92} - 1267 q^{94} + 1057 q^{96}+O(q^{100}) \) Copy content Toggle raw display

Display 100 coefficients 10 coefficients

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{7} - 14x^{5} + 56x^{3} - 56x - 15 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 4 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( \nu^{3} - 6\nu \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( \nu^{4} - \nu^{3} - 8\nu^{2} + 6\nu + 8 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( \nu^{5} - 10\nu^{3} - 2\nu^{2} + 20\nu + 8 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( \nu^{6} - 12\nu^{4} + 36\nu^{2} - 4\nu - 16 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 4 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{3} + 6\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{4} + \beta_{3} + 8\beta_{2} + 24 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( \beta_{5} + 10\beta_{3} + 2\beta_{2} + 40\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( \beta_{6} + 12\beta_{4} + 12\beta_{3} + 60\beta_{2} + 4\beta _1 + 160 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/287\mathbb{Z}\right)^\times\).

\(n\) \(206\) \(211\)
\(\chi(n)\) \(-1\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

comment: embeddings in the coefficient field
 
gp: mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
286.1
−0.292301
−0.957283
1.48399
2.01727
−2.38176
−2.67770
2.80779
−3.91456 −5.59495 11.3238 0 21.9018 7.00000 −28.6694 22.3034 0
286.2 −3.08361 3.35781 5.50864 0 −10.3542 7.00000 −4.65206 2.27491 0
286.3 −1.79777 −0.867827 −0.768030 0 1.56015 7.00000 8.57181 −8.24688 0
286.4 0.0693632 4.10058 −3.99519 0 0.284429 7.00000 −0.554572 7.81476 0
286.5 1.67278 5.98117 −1.20181 0 10.0052 7.00000 −8.70148 26.7744 0
286.6 3.17010 −5.18274 6.04955 0 −16.4298 7.00000 6.49729 17.8608 0
286.7 3.88369 −1.79404 11.0831 0 −6.96751 7.00000 27.5084 −5.78141 0
\(n\): e.g. 2-40 or 990-1000
Embeddings: e.g. 1-3 or 286.7
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
287.d odd 2 1 CM by \(\Q(\sqrt{-287}) \)

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 287.3.d.b yes 7
7.b odd 2 1 287.3.d.a ✓ 7
41.b even 2 1 287.3.d.a ✓ 7
287.d odd 2 1 CM 287.3.d.b yes 7
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
287.3.d.a ✓ 7 7.b odd 2 1
287.3.d.a ✓ 7 41.b even 2 1
287.3.d.b yes 7 1.a even 1 1 trivial
287.3.d.b yes 7 287.d odd 2 1 CM

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{3}^{\mathrm{new}}(287, [\chi])\):

\( T_{2}^{7} - 28T_{2}^{5} + 224T_{2}^{3} - 448T_{2} + 31 \) Copy content Toggle raw display
\( T_{3}^{7} - 63T_{3}^{5} + 1134T_{3}^{3} - 5103T_{3} - 3718 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{7} - 28 T^{5} + \cdots + 31 \) Copy content Toggle raw display
$3$ \( T^{7} - 63 T^{5} + \cdots - 3718 \) Copy content Toggle raw display
$5$ \( T^{7} \) Copy content Toggle raw display
$7$ \( (T - 7)^{7} \) Copy content Toggle raw display
$11$ \( T^{7} \) Copy content Toggle raw display
$13$ \( T^{7} - 1183 T^{5} + \cdots - 24950438 \) Copy content Toggle raw display
$17$ \( T^{7} - 2023 T^{5} + \cdots - 800009246 \) Copy content Toggle raw display
$19$ \( T^{7} - 2527 T^{5} + \cdots - 305833574 \) Copy content Toggle raw display
$23$ \( T^{7} + \cdots - 6483516722 \) Copy content Toggle raw display
$29$ \( T^{7} \) Copy content Toggle raw display
$31$ \( T^{7} \) Copy content Toggle raw display
$37$ \( T^{7} + \cdots - 101889625898 \) Copy content Toggle raw display
$41$ \( (T + 41)^{7} \) Copy content Toggle raw display
$43$ \( T^{7} + \cdots - 358855177802 \) Copy content Toggle raw display
$47$ \( T^{7} + \cdots + 987930187618 \) Copy content Toggle raw display
$53$ \( T^{7} \) Copy content Toggle raw display
$59$ \( T^{7} \) Copy content Toggle raw display
$61$ \( T^{7} \) Copy content Toggle raw display
$67$ \( T^{7} \) Copy content Toggle raw display
$71$ \( T^{7} \) Copy content Toggle raw display
$73$ \( T^{7} \) Copy content Toggle raw display
$79$ \( T^{7} \) Copy content Toggle raw display
$83$ \( T^{7} \) Copy content Toggle raw display
$89$ \( T^{7} + \cdots + 87666289649746 \) Copy content Toggle raw display
$97$ \( T^{7} + \cdots + 74677982531458 \) Copy content Toggle raw display
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